Yes they can grow out of it. I think it varies with each child, but I would say about 2 years if they are not allergic to it. When we celebrated my sons christening, I gave a teaspoon of pavlova not thinking about the egg. A few minutes later he came up in a rash around his mouth. He wasn’t complaining about it, which was good.
I tried him on scrambled egg every 4-6 mths to see if he was still having a reaction and by the time he was 2 he had grown out of it! Scrambled eggs is one of his favourite dishes.